Course FM/2 Practice Exam 1 Solutios Solutio 1 D Sikig fud loa The aual service paymet to the leder is the aual effective iterest rate times the loa balace: SP X 0.075 To determie the aual sikig fud paymet, we first eed to determie the relevat accumulated value factor: 5 (1.06) 1 s 56% 5.63709 0.06 The aual sikig fud paymet is: X X SFP s 56% 5.63709 The total aual paymet from the borrower is the combiatio of the service paymet ad the sikig fud paymet, so the total paymets made by Lace over the 5-year period is just five times this amout. This give us a equatio of value, which we ca solve for the loa amout, X: X 17,036.75 X(0.075)(5) (5) 5.63709 17,036.75 0.375X 0.88698X 17,036.75 1.26198X X 13,500.00 Solutio 2 C Iterest rate swap The otioal amout is ot eeded to aswer this questio. The zero-coupo bod prices are: 1 P(0,1) 1.045 0.956938 2 P(0,2) 1.048 0.910495 3 P(0,3) 1.051 0.861374 ActuarialBrew.com 2013 Page 1
The 1-year implied forward rates are: r (0,1) s 0.045 0 0 2 1.048 r0 (1,2) 1 0.051009 1.045 3 1.051 r0 (2,3) 1 0.057026 2 1.048 The fixed swap rate is: P(0,1) r 0(0,1) P(0,2) r0(1,2) P(0,3) r0(2,3) R P(0,1) P(0,2) P(0,3) 0.956938(0.045) 0.893364(0.051009) 0.861374(0.057026) 0.956938 0.910495 0.861374 0.138626 0.0508 2.728807 A quicker way to determie the aswer is: 3 1 P(0,3) 1 1.051 R 0.0508 P (0,1) P (0,2) P (0,3) 2.728807 Solutio 3 A Accumulated value of cash flows at aual effective iterest rate The equatio of value for Sheryl s accumulated value at time 10 years is: 10 5 0 39,661.45 5,000(1 i) 10,000(1 i) 15,000(1 i) 5 We let x (1 i), ad we ca solve the equatio of value for x with the quadratic equatio: 2 0 5,000x 10,000x 24,661.45 x x 2 10,000 (10,000) 4(5,000)( 24,661.45) 2(5,000) 10,000 24,356.2928 10,000 We discard the egative solutio sice it does t make sese with iterest rates, ad the we solve for i: x 1.43563 5 (1 i) 1.43563 1 i 1.075 i 0.075 ActuarialBrew.com 2013 Page 2
Solutio 4 B Varyig perpetuity-immediate ad level perpetuity-due Bob s preset value is: 50 50(1 i) 50a i/(1 i) i Tom s paymets are a compoud-icreasig perpetuity-immediate. The preset value of which is: a j lim lim 1 1 (1 j) 1 1 a j 1 e j 1 e j Sice j (1 e) /( i e), we have: a j 1 1 1 1 e 1 1e j 1 e i e ie Tom s preset value is: 5 5a j i 0.025 Sice their preset values are equal, we ca set up the equatio of value ad solve for i: 50(1 i) 5 i i 0.025 50(1 i)( i0.025) 5i 2 50( i0.025 i 0.025 i) 5i 2 50i 43.75i1.25 0 2 43.75 43.75 4(50)( 1.25) i 2(50) 2.76949 i 100 i 0.0277 We discarded the egative solutio sice it did t make sese for a iterest rate. ActuarialBrew.com 2013 Page 3
Solutio 5 B Risk maagemet: sell collar AA sells a collar by sellig a lower-strike put optio ad buyig a higher-strike call optio, so AA sells the $2.45 strike put ad buys the $2.55 strike call. AA receives $0.24 for sellig the put ad pays $0.08 for buyig the call. The accumulated cost of the hedge i oe year is egative: 0.05351 ( 0.24 0.08) e 0.16879 I other words, the hedge produces positive cash flow at time 0, which is accumulated to time oe year. If the price of jet fuel i oe year is $2.50, the payoffs of the put ad call are: Put payoff Max[0,2.45 2.50] 0 Call payoff Max[0,2.50 2.55] 0 So both the put ad the call are out of the moey. If the price of jet fuel i oe year is $2.50, the profit at that time o oe gallo is the reveue less the expese plus the payoffs of the put ad call plus the accumulated proceeds from settig up the hedge at time 0: 2.702.50000.16879 0.36879 If the price of jet fuel i oe year is $2.75, the payoffs of the put ad call are: Put payoff Max[0,2.45 2.75] 0 Call payoff Max[0,2.75 2.55] 0.20 The put is out of the moey but the call is i of the moey. If the price of jet fuel i oe year is $2.75, the profit at that time o a per gallo basis is: 2.70 2.75 0 0.20 0.16879 0.31879 Solutio 6 A Callable bod price Let s work i semiaual effective periods ad defie the bod variables first: F C 1,000 30 2 60 i 0.08 / 2 0.04 g r 0.06 / 2 0.03 coupo 0.03 1,000 30 Sice g < i, the bod is a discout bod, ad the miimum yield is determied from a call at the latest possible call date. (If the call price chages, the we would eed to check the price at the latest date of each call price chage.) The latest call date i this case is to assume that the bod will mature at time 30 years. ActuarialBrew.com 2013 Page 4
The appropriate auity factor is: a 60 4% 60 1 (1.04) 22.62349 0.04 The price of the bod that guaratees a miimum yield of at least a 8% omial aual rate covertible semiaually is: 60 4% 60 P 30a 1,000(1.04) 773.7651 Let s check this aswer usig the BA-35 calculator by pressig [2d][CMR], 1,000 [FV], 30 [PMT], 4 [%i], 60 [N], ad [CPT][PV], ad the result is 773.7651. Usig the BA II Plus, we press [2d][CLR TVM], 1,000 [FV], 30 [PMT], 4 [I/Y], 60 [N], ad [CPT][PV], ad the result is the same. We ca verify that this is the price to guaratee the required miimum retur by calculatig the prices if the bod had bee called at other dates. If the prices at other possible call dates are higher, the the retur would be lower. The table below illustrates the prices at other chose call dates: N Price 40 802.0723 41 800.0695 58 775.7043 59 774.7157 Sice all the other prices are higher tha $773.7651, they would provide a lower retur tha the required retur if the bod is ot called early. Solutio 7 C Nomial iterest rate ad force of iterest The accumulated value of Peter s accout after 6.5 years is: 46.5 (4) i X 1 4 The accumulated value i David s accout after 6.5 years is: 0.076.5 X e ActuarialBrew.com 2013 Page 5
Sice their accumulated values are equal at this time, we set up the equatio of value ad solve for the aual omial rate of iterest covertible quarterly: 26 (4) i 0.455 X 1 Xe 4 26 (4) i 1 1.576173 4 (4) i 1 1.01765 4 (4) i 0.07062 Alteratively, it is quicker if we ote that both X ad 6.5 years are ot ecessary to work this problem. We the have: 4 (4) i 0.07 1 e 4 (4) i 0.0706 Solutio 8 E Bod coupo Let s work i semiaual effective periods ad defie the bod variables first: F 1,000 C 1,050 30 2 60 i 0.08 / 2 0.04 60 1 (1.04) a 22.62349 60 4% 0.04 We set up the equatio of value for the price of the bod ad we ca solve it for X, the coupo amout: 60 948.19 Xa 1,050(1.04) 60 4% 948.19 22.62349X 99.81342 X 37.50 The semiaual effective coupo rate is 37.50 /1,000 3.75%, ad the aual coupo rate covertible semiaually is twice this amout: 3.75% 2 7.5%. ActuarialBrew.com 2013 Page 6
Alteratively, we ca use a fiacial calculator to help us determie the aswer. Usig the BA 35, we press [2d][CMR], 948.19 [PV], 4 [%I], 1,050 [FV], 60 [N], [CPT][PMT], ad the result is 37.50. Usig the BA II Plus, we press [2d][CLR TVM], 948.19 [PV], 4 [I/Y], 1,050 [FV], 60 [N], [CPT][PMT], ad we get the same result. Solutio 9 C Macaulay duratio of stock with level divideds The formula for Macaulay duratio is: mt y tcft 1 m MacD mt y CFt 1 m The deomiator i this case is just the preset value of a level perpetuity-immediate, which is give by: 2 3 2 1v v 1/(1 i) 1i 1 1v1v 1v v1vv v 1v 1v 11/(1 i) i(1 i) i The umerator is the preset value of a icreasig perpetuity-immediate: a v a 0 1/ 1 ( ) lim ( ) lim d i Ia Ia i i i 2 i We the have: 2 (1 i) / i 1 i MacD 1/ i i We are give tha the Macaulay duratio is 21.0, so we ca solve the equatio for i: 1 i 21.0 i 21.0i 1 i 20.0i 1 i 0.05 ActuarialBrew.com 2013 Page 7
Solutio 10 A Put-call parity Usig put-call parity, we solve for the ukow cotiuously compouded iterest rate r: Call( K, T ) Put( K, T ) S0 PV ( K ) 0.5r 7.69 9.39 120 125e 0.5r 121.70 125e 0.5r 0.9736 e l(0.9736) r 0.50 r 0.0535 Solutio 11 D Pthly auity-due preset value Liz s auity paymets occur today, ad at times 2 years, 4 years, 6 years, ad so o, util at time 18 years. There are 10 paymets i all. Sice these paymets occur every other year, let s work i two-year effective periods. If we let i deote the two-year effective iterest rate, the equatio of value is: 10 1 (1 i) 62,787.98 10,000a 10,000 10 i/(1 i) We could use trial ad error to determie the value of i that satisfies the above equatio. It is a valid approach durig the exam if it saves time, but we ca also use a calculator to quickly determie the aswer. Usig the BA 35, we press [2d][CMR], [2d][BGN], 10,000 [PMT], 10 [N], 68,752.84 [PV], [CPT][%i], ad the result is 9.50%. Usig the BA II Plus, we press [2d][CLR TVM], [2d][BGN] [2d][SET] [2d][QUIT], 10,000 [PMT], 10 [N], 68,752.84 [PV], [CPT][I/Y] ad we get the same result. We eed to be careful, sice this rate of 9.5% is the two-year effective iterest rate. We eed to covert this rate to the aual omial iterest rate covertible semiaually: 1/2 (2) (1/2) i i 0.5 1 1 (1.095) 1 0.02295 2 1/2 (2) i 0.04590 ActuarialBrew.com 2013 Page 8
Solutio 12 E Forward iterest rate The aswer ca be quickly determied by the followig relatioship: 4 4 (1 s4 ) 1.09 f3 1 1 0.12055 3 3 (1 s3 ) 1.08 Alteratively, sice forward rates ca be a little cofusig, it may help to draw a timelie. The timelie below shows the various spot ad forward rates ad where they are placed o the timelie. f0 f1 f2 f3 0 1 2 3 4 s1 = 6.0% s2 = 7.0% s3 = 8.0% s4 = 9.0% The oe-year forward rate coverig the spa of the fourth year from time 3 years to time 4 years is f 3. To determie this forward rate, we eed to calculate f 1 ad f 2 first: f0 s1 6.0% 2 1.07 f1 1 0.08009 1.06 3 1.08 f2 1 0.10028 (1.06)(1.008009) 4 1.09 f3 1 0.12055 (1.06)(1.08009)(1.10028) Solutio 13 B Dollar-weighted iterest rate The fud balace of $1,050 at the start of the year is accumulated with 12 moths of iterest. The withdrawal o May 1 is accumulated with 8 moths of iterest ad the withdrawal o Jue 15 is accumulated with 6.5 moths of iterest. There are deposits at the ed of every moth from Jauary 31 to December 31, ad each of these deposits are accumulated to the ed of the year. The fud value at the ed of the year is $1,160. ActuarialBrew.com 2013 Page 9
We let i be the aual effective iterest rate, ad the equatio of value is: 12/12 11/12 10/12 1/12 0/12 1,050(1 i) 90(1 i) 90(1 i) 90(1 i) 90(1 i) 8 /12 6.5 /12 600(1 i) 400(1 i) 1,160 Sice the fud activity occurs durig a 12-moth period, we ca use the simple iterest approximatio to simplify the above equatio: 12 11 10 1 0 1,050(1 i) [90(1 i) 90(1 i) 90(1 i) 90(1 i)] 12 12 12 12 12 8 6.5 600(1 i) 400(1 i) 1,160 12 12 Recogizig that 12 ( )( 1)/2, the part i the brackets above becomes: 90 90 90 12 i(11 10 1) 90 12 i(66) 1,080 495i 12 12 We ca ow solve the full equatio for i, the aual effective iterest rate: 1,050 1,050i1,080 495i600 400i400 216.6667i 1,160 928.3333i 30 i 0.0323 Lastly, we solve for the omial aual rate compouded mothly: i 12 (1.0323) 1 0.0318 (12) 1 /12 Solutio 14 A Loa: amout of pricipal i a paymet The loa paymet P is the sum of the iterest compoet ad the pricipal compoet: P 2,058.08 297.79 2,355.87 We ca determie the balace at time 12 years from the iterest compoet of the 13th aual paymet: I13 B12 i 2,058.08 B12 22,867.55556 0.09 The balace at time 12 years is also equal to the preset value of the remaiig future loa paymets. There are 12 remaiig loa paymets, so aother equatio for the loa balace at time 12 years is: B 12 a 12 9% 2,355.87 22,867.55556 Usig the BA 35 calculator, we press [2d][CMR], 22,867.55556 [PV], 9 [%i], 2,355.87 [PMT], [CPT][N], ad the result is 24.00014. Usig the BA II Plus, we press [2d][CLR TVM], 22,867.55556 [PV], 9 [I/Y], 2,355.87 [PMT], [CPT][N] ad we get the same result. ActuarialBrew.com 2013 Page 10
Sice 12 24, we kow 36, i.e., the loa iitially had 36 aual paymets. To calculate the amout of pricipal i the 23rd paymet, we eed to kow the balace at time 22 years. At time 22 years, there are 36 22 14 paymets remaiig, so the appropriate auity factor is: a 14 9% 14 1 (1.09) 7.78615 0.09 The balace at time 22 years is the: B 2,355.87a 18,343.15812 22 14 9% Now we ca determie the amout of iterest ad pricipal i the 23rd paymet: I23 ib22 0.09 18,343.15812 1,650.88423 P23 P I23 2,355.87 1,650.88423 704.98577 Solutio 15 A Auity-immediate preset value With iformatio from auity 2, we ca determie the aual effective iterest rate. Usig the BA 35, we press [2d][CMR], 475.54 [FV], 5 [N], 81.06 [PMT], [CPT][%i], ad the result is 7.9993, or 8.0%. Usig the BA II Plus or the BA II Plus Professioal, we press [2d][CLR TVM], 475.54 [FV], 5 [N], 81.06 [PMT], [CPT][I/Y] ad we get the same result. Sice the accumulated value of auity 2 at time 5 years is $475.54, the preset value at time 0 of auity 2 is: PV 5 (1.08) (475.54) 323.64453 Auity 1 has five aual paymets from time 6 years to time 10 years. If we use a auity-immediate preset value factor to value these paymets, its value is determied at time 5 years (1 year before the first paymet), so we eed to discout the time 5 preset value back 5 years to determie the time 0 preset value. Sice the preset value of auity 1 is twice the preset value of auity 2, we set up the equatio of value for auity 1 ad solve for the ukow paymet X: 5 2(323.64453) X(1.08) a 58% 5 5 1 (1.08) 647.28907 X(1.08) 0.08 2.71737X 647.28907 X 238.20 ActuarialBrew.com 2013 Page 11
Solutio 16 C Short sale Sice we kow all of the iformatio for Steve, we ca calculate his yield: S B (margi i)(margi req %)( S) div SS yield (margi req %)( S) (500 475) (0.06)(0.75)(500) 12.5 (0.75)(500) 0.09333 Sice Laura s short sale yield is the same as Steve s, we ca solve for her buy back price. X: 0.09333 617 X 0.09333 450 X 575 600 X (0.06)(0.75)(600) 10.0 (0.75)(600) Solutio 17 E Bod yield: reivestmet of coupo paymets Let s work i semiaual effective periods ad defie the bod variables first: F 1,000 C 1,050 15 2 30 i 0.08 / 2 0.04 r 0.07 / 2 0.035 coupo 0.035 1,000 35 a 30 4% 1 (1.04) 0.04 The price of the bod is: 30 4% 30 17.29203 30 P 35a 1,050(1.04) 928.95577 Alteratively, usig the BA 35, we press [2d][CMR], 1,050 [FV], 30 [N], 35 [PMT], 4 [%i], [CPT][PV], ad the result is 928.95577. Usig the BA II Plus, we press [2d][CLR TVM], 1,050 [FV], 30 [N], 35 [PMT], 4 [I/Y], [CPT][PV], ad we get the same result. ActuarialBrew.com 2013 Page 12
George reivests each of the 30 coupos at a semiaual effective iterest rate of 0.09 / 2 0.045. The accumulated value of these coupos at time 15 years is: 30 (1.045) 1 35s 35 2,135.24744 30 4.5% 0.045 To determie his aual effective yield, we recogize that George paid $928.95577 at time 0, ad it grew to $1,050 + $2,135.24744 = 3,185.24744 at time 15 years. We set up this equatio of value ad solve for i, the aual effective iterest rate: 15 928.95577(1 i) 3,185.24744 15 (1 i) 3.42885 1 i 1.08562 i 0.08562 Solutio 18 D Sythetic log stock The payoff graph for a log stock positio looks like this: Log stock payoff Combiig the payoffs of the log forward ad the log zero-coupo bod (i.e., cash positio), we get the same payoff patter as the log stock. Log forward Log zero-coupo bod Log stock + = As a aside, we ote that aswer choice A produces the same cash flows as a short sale of a stock. ActuarialBrew.com 2013 Page 13
Solutio 19 E Futures cotract ad margis The otioal value of oe S&P 500 cotract is defied to be $250 times the value of the S&P idex. This is the stadard assumptio ad should be used uless otherwise defied i the questio. The otioal value of 5 S&P 500 cotracts is: 5 250 1,500 1,875,000 The iitial margi is 20% of this amout: 0.20 1,875,000 375,000 After oe week, the iitial margi has accumulated to: 0.04 (1/ 52) 375,000e 375,288.5725 We receive a margi call whe the margi falls to 75% of the iitial margi: 0.75 375,000 281,250.00 The 5 log S&P futures cotracts require us to pay 5 250 1,250 times the forward price at expiratio of the futures cotract. The equatio of value becomes: 375,288.5725 ( F1 W 1,500) 1,250 281,250.00 ( F1 W 1,500) 1,250 94,038.5725 F1 W F1 W 1,500 75.23086 1,424.7691 This is the greatest S&P 500 idex futures price at which we receive a margi call. Solutio 20 B Auity-due accumulated value If Joh had made all of the aual deposits from 1/1/78 to 12/31/07, he would have made 2007 1978 1 30 deposits. The deposits were supposed to occur o Jauary 1 of each year from 1/1/78 to 1/1/07. Joh missed deposits 16 through 21. The 16th deposit was supposed to occur o 1/1/93, sice the first deposit occurred o 1/1/78. Likewise, the 21st deposit was supposed to occur o 1/1/98. Thus, Joh made the first 15 deposits (from 1/1/78 to 1/1/92), missed the ext 6 deposits (from 1/1/93 to 1/1/98), ad made the last 9 deposits (from 1/1/99 to 1/1/07). The accumulated value of the first 15 deposits at time 16 years (1/1/93) is: 15 (1.045) 1 10,000 s 10,000 217,193.3673 15 4.5% 0.045 /1.045 ActuarialBrew.com 2013 Page 14
Sice we eed the accumulated value as of 12/31/07 (or 1/1/08), we eed to accumulate this accumulated value for aother 2008 1993 15 years. The accumulated value of the first 15 deposits at time 12/31/07 is: 15 217,193.3673(1.045) 420,330.5106 The accumulated value of the last 9 deposits at 12/31/07 (i.e., 9 years after 1/1/99) is: 9 (1.045) 1 10,000 s 10,000 112,882.0937 94.5% 0.045 /1.045 The total accumulated value of these two pieces at time 1/1/08 is: 420,330.5106 112,882.0937 533,212.6043 Alteratively, we ca approach this problem from aother agle. We ca assume that Joh made all 30 paymets. The accumulated value of the 30 aual paymets at 12/31/07 is: 30 (1.045) 1 10,000 s 10,000 637,523.8779 30 4.5% 0.045 /1.045 The accumulated value of the 6 deposits that were missed, valued at 1/1/99 is: 6 (1.045) 1 10,000 s 10,000 70,191.5179 64.5% 0.045 /1.045 We eed to accumulate the 1/1/99 accumulated value of the missig deposits to 12/31/07. There are 2007 1999 1 9 years from 1/1/99 to 12/31/07, so the 12/31/07 accumulated value of the missed deposits is: 9 70,191.5179(1.045) 104,311.2736 Now we ca subtract the accumulated value at 12/31/07 of the missig deposits from the accumulated value at 12/31/07 of the deposits assumig that oe were missed. The resultig accumulated value is: 637,523.8779 104,311.2736 533,212.6043 Solutio 21 C Macaulay duratio of bod The formula for Macaulay duratio is: MacD mt y tcft 1 m mt y CFt 1 m ActuarialBrew.com 2013 Page 15
For this bod with coupo paymets of $X at the ed of each year, the Macaulay duratio is: 1 2 3 10 10 X(1)(1.09) X(2)(1.09) X(3)(1.09) X(10)(1.09) 400(10)(1.09) 1 2 3 10 10 X(1.09) X(1.09) X(1.09) X(1.09) 400(1.09) This ca be writte as: 7.466 10 X( Ia) 400(10)(1.09) 10 9% 10 Xa 400(1.09) 10 9% Determiig the required values, we have: 10 1 (1.09) a 6.41766 10 9% 0.09 a (1.09)(6.41766) 6.99525 10 9% 10 6.99525 10(1.09) ( Ia) 30.79043 10 9% 0.09 Pluggig these values ito the above equatio, we ca solve for X: X(30.79043) 1,689.64323 7.466 X(6.41766) 168.96432 47.91423X 1,261.48763 30.79043X 1,689.64323 17.12380X 428.15559 X 25.00 Solutio 22 B Compoud decreasig ad icreasig auity The preset value of a compoud icreasig auity-immediate is: 1 1 1 (1 j) ie a where j 1 e j 1e j 1e Sice we have mothly paymets ad icreases that occur mothly, let s work i mothly periods. The mothly effective iterest rate is: (12) i 1/12 (1.125) 1 0.009864 12 Let s split the auity ito two parts: the compoud decreasig part durig the first two years ad the compoud icreasig part durig the last two years. The mothly paymets for the compoud decreasig part start at time 1 moth. The first paymet is $5,000. There are 24 paymets for the decreasig part ad the mothly rate of decrease is 1%. ActuarialBrew.com 2013 Page 16
The mothly iterest rate for the first two years is: 0.009864 ( 0.01) j 0.020064 1 ( 0.01) The preset value at time 0 of the decreasig part is: 24 1 1 (1.020064) 5,000 95,455.41662 1 ( 0.01) 0.020064 The mothly paymets for the compoud icreasig part start at time 25 moths. The 23 first paymet for the icreasig part is 5,000(0.99) (1.005). There are also 24 paymets for the icreasig part ad the mothly rate of icrease is 0.5%. The mothly iterest rate for the secod two years is: 0.009864 0.005 j 0.004839 1.005 The preset value at time 24 moths of the icreasig part is: 24 23 1 1 (1.004839) 5,000(0.99) (1.005) 89,706.74436 1 (0.005) 0.004839 The preset value of the decreasig part must be accumulated for five years ad the preset value of the icreasig part must be accumulated for three years to determie the accumulated value of the etire series of paymets at time five years: 5 3 95,455.41662(1.125) 89,706.74436(1.125) 299,740.7459 Solutio 23 E Pricig a bod usig spot rates The bod pays aual coupos of 0.08 1,000 80. The preset value of the bod is: 80 80 80 1,080 PV 973.95260 1.06 2 3 4 1.07 1.08 1.09 The aual effective yield ca be quickly determied usig a fiacial calculator. Usig the BA 35, we press [2d][CMR], 1,000 [FV], 4 [N], 80 [PMT], 973.9526 [PV], [CPT][%i], ad the result is 8.80048. Usig the BA II Plus, we press [2d][CLR TVM], 1,000 [FV], 4 [N], 80 [PMT], 973.9526 [PV], [CPT][%i], ad we get the same result of 8.8%. ActuarialBrew.com 2013 Page 17
Solutio 24 D Decreasig auity-immediate preset value The paymets start at $1,050 at time 1 year ad decrease by $15 each year. There are 30 paymets, so the last paymet of $615 occurs at time 30 years. The decreases are $15 ad there are 30 paymets, so let s subtract 15 30 450 from the first paymet of $1,050, ad we re left with $600. If we subtract $600 from each of the 30 paymets, we are left with a paymet stream that starts at $450 at time 1 ad decreases by $15 each year to $15 at time 30. The preset value of a level series of 30 paymets of $600, we have: 30 1 1.065 600a 600 600(13.05868) 7,835.20554 30 6.5% 0.065 The preset value of the decreasig paymet stream is: 30 a 30 6.5% 30 13.05868 15( Da) 15 15 3,909.53633 30 6.5% 0.065 0.065 Puttig the two preset values back together, we have: PV 7,835.20554 3,909.53633 11,744.7419 Solutio 25 A Loa balace We are give that the outstadig loa balace at the ed of the 9th year is $1,355.22. So the fial loa paymet must be this balace times the quatity of oe plus the effective iterest rate, sice the fial paymet must pay off this balace plus the iterest from the last period o this balace. The level paymet is the: P 1,355.22 1.075 1,456.8615 The balace at the ed of the third year is the preset value of the remaiig seve paymets: B 7 1 (1.075) 1,456.8615a 1,456.8615 7,716.41 0.075 3 77.5% Alteratively, sice this is a 10-year loa, the fial loa paymet must cotai exactly this amout as the pricipal amout to pay off the loa, so: P10 1,355.22 ActuarialBrew.com 2013 Page 18
We ca develop a relatioship betwee successive pricipal amouts. We recall that the outstadig loa balace at ay time t is the preset value of the remaiig loa paymets: t 1 v Bt Pa P ti i The amout of iterest i loa paymet t is the periodic effective iterest rate times the prior loa balace, ad the amout of pricipal i loa paymet t is the loa paymet mius the amout of iterest i loa paymet t: t 1 1 v t1 It ibt1 ip P 1 v i t1 t1 Pt P It P P 1 v Pv So if we divide successive loa paymets, we see that a pricipal amout i a loa paymet is (1 + i) times the pricipal amout i the prior loa paymet: Pt 1 Pt t Pv 1 1 i t 1 Pv v The iitial loa amout L is the sum of the pricipal amouts i each loa paymet. I this case, we have: L P1 P2 P10 We have just determied a iterative relatioship betwee the pricipal amouts i each loa paymet, so we ca restate the above relatioship as: 9 8 L 1,355.22v 1,355.22v 1,355.22 2 9 1,355.22[1 vv v ] 10 1 v 1,355.22 1 v 10 1 (1.075) 1,355.22 1 (1.075) 1 10,000.0153 So the loa balace at the ed of the third year is the iitial loa amout less the pricipal amouts i the first three loa paymets: B3 LP1 P2 P3 9 8 7 10,000.0153 1,355.22(1.075) 1,355.22(1.075) 1,355.22(1.075) 7,716.41 ActuarialBrew.com 2013 Page 19
Solutio 26 B Net preset value The et preset values for projects X ad Y are: 5 10 NPV 1,000 700(1 i) 700(1 i) X 5 10 NPV 800(1 i) 1,016.31(1 i) Y Sice their et preset values are equal, we set up the equatio of value. We let 5 i x (1 ) ad solve for x usig the quadratic equatio: 5 10 5 10 1,000 700(1 i) 700(1 i) 800(1 i) 1,016.31(1 i) 2 316.31x 1,500x 1,000 0 2 1,500 ( 1,500) 4(316.31)(1,000) x 2(316.31) x 0.80246 or 3.93973 We ca ow solve for the ukow aual effective iterest rate: 5 (1 i) 0.80246 or 5 (1 i) 3.93973 1 i 1.0450 1 i 0.76016 i 0.045 i 0.23984 Sice the egative iterest rate does t make sese, the aswer is 4.5%. Solutio 27 A Surplus The preset value of the liability is: 5,000,000 PVL 2,086,325.304 15 1.06 The compay eeds to buy a bod that will mature for the liability amout of $5,000,000 at time 15 years, assumig the iterest rate does ot chage. Sice the coupo rate of the 15-year bod is equal to its yield, the 15-year bod is priced at par. The bod s face amout should therefore also be $2,086,325.304. To verify this, we have: bod F 2,086,325.304 aual coupo 0.06 2,086,325.304 125,179.5182 ActuarialBrew.com 2013 Page 20
This bod will exactly match the liability i 15 years, sice the bod s face amout plus the reivested coupos will add to $5,000,000 at time 15 years, assumig the iterest rate does ot chage: 2,086,325.304 125,179.51852s 2,086,325.304 125,179.5182 15 6% 5,000,000.00 15 1.06 1 The iterest rate chages o 12/31/10, which is exactly 3 years after 12/31/07. The ew iterest rate of 5.5% remais i effect for the remaiig 12 years util 12/31/22. The coupos are reivested for 3 years at 6.0%, ad the they are reivested for the remaiig 12 years at 5.5%. The accumulated value of the coupos is the: 0.06 12 AV (coupos) 15 125,179.5182 s (1.055) 125,179.5182s 36% 125.5% 3 12 1.06 1 12 1.055 1 125,179.5182 (1.055) 125,179.5182 0.06 0.055 2,808,812.43 Combied with the face amout, the bod s value at 12/31/22 is the: 2,808,812.43 2,086,325.30 4,895,137.73 The isurace compay s liability at 12/31/22 is still $5,000,000, so the isurace compay s profit at this time is: 4,895,137.73 5,000,000 104,862.27 Solutio 28 B Level auity-due accumulated value factor Expressio A is true sice: (1 i) 1 (1 i) 1 s sice d iv d iv Expressio B is false sice: 1 s (1 i) a (1 i) (1 i) a (1 i) a Expressio C is true sice: 1v 1v s (1 i) a (1 i) (1 i) sice d 1 v d 1 v ActuarialBrew.com 2013 Page 21
Expressio D is true sice: 1 2 1 s (1 i) (1 i) (1 i) (1 i) Expressio E is true sice: s s 1 1 1 (1 i) 1 (1 i) 1 s (1 i) (1 i) so s s 1 Solutio 29 C Callable bod price Workig with what we have, we ca quickly determie the bod s aual coupo paymet $C. The we calculate the price that results i a aual effective yield of 8.34%. There is o eed to determie P. To determie $C, we assume the bod is called at the ed of the 15th year. Usig the BA 35, we press [2d][CMR], 1,025 [FV], 15 [N], 8 [%i], 793.89 [PV], [CPT][PMT], ad the result is 55.00. Usig the BA II Plus, we press [2d][CLR TVM], 1,025 [FV], 15 [N], 8 [I/Y], 793.89 [PV], [CPT][PMT], ad we get the same result. The ivestor actually held the bod for 20 years, whe it was called for $1,025, ad the ivestor s actual aual effective yield was 8.34%. The price that results i a aual effective yield of 8.34% ca ow be determied. Usig the BA 35, we press [2d][CMR], 1,025 [FV], 20 [N], 55 [PMT], 8.34 [%i], [CPT][PV], ad the result is 733.1175. Usig the BA II Plus, we press [2d][CLR TVM], 1,025 [FV], 20 [N], 55 [PMT], 8.34 [I/Y], [CPT][PV], ad we get the same result. Thus, we coclude that the ivestor paid $733.12. Solutio 30 D Mothly auity-immediate preset value Sice Tia s retiremet paymets occur mothly, let s work i mothly periods. We set up the equatio of value for the preset value of these beefits, ad we will be able to determie the aual effective iterest rate. Tia will receive 240 mothly paymets at the ed of each moth for 20 years. 4,000a 587,938.54 240 ActuarialBrew.com 2013 Page 22
A fiacial calculator ca quickly determie the iterest rate. Sice we re workig i moths, the result will be a mothly effective iterest rate, which we ca covert to a aual effective iterest rate. Usig the BA 35, we press [2d][CMR], 240 [N], 4,000 [PMT], 587,938.54 [PV], [CPT][%i], ad the result is 0.44717. Usig the BA II Plus, we press [2d][CLR TVM], 240 [N], 4,000 [PMT], 587,938.54 [PV], [CPT][%i], ad we get the same result. The aual effective iterest rate is: 12 i (1.0047717) 1 0.0550 The preset value of the zero-coupo bod is the: 1,000,000 X 381,465.904 18 1.055 Solutio 31 B Prepaid forward, cost of carry The cost of carry is the cotiuously compouded risk-free iterest rate less the cotiuously paid divided rate. We are give the latter, so we eed to calculate the former to determie the cost of carry. I this case, we re give the forward premium but ot the forward price. We do ot eed the forward premium to calculate the forward price i this problem. We recall the forward premium is just the forward price divided by the stock price. Usig this fact, we ca modify the formula for the forward price to determie the cost of carry r without specifically solvig for r: F F ( r ) T 0, T S0e 0, T ( r ) T e 0 ( r )0.75 S 1.054 e l(1.054) r 0.75 r 0.0701 Solutio 32 C Immuizatio Statemet I is ot true sice Redigto (i.e., classical) immuizatio oly protects a portfolio agaist small chages i iterest rates. Statemet II is ot true sice the duratio of the assets must be established so that it approximately matches the duratio of the liabilities i order to meet the secod immuizatio coditio. ActuarialBrew.com 2013 Page 23
Statemet III is true sice the covexity of the assets must be greater tha the covexity of the liabilities i order to meet the third immuizatio coditio of Redigto immuizatio. Oly statemet III is true, so choice C is the correct aswer. Solutio 33 D Dollar-weighted ad time-weighted iterest rates Let s make a table of the give iformatio, where the fud values are valued immediately before the ext cash flow occurs. t Ft ct 0 100 0 0.5 90 20 0.75 105 40 1.0 X 0 The equatio for the time-weighted iterest rate, i, is: 1 90 105 X (1 i) 100 90 20 105 40 1 i 0.009310X We are give that the time-weighted rate, i, equals the dollar-weighted rate, j, plus 0.097. Substitutig this ito the above equatio, we have: 1 ( j0.097) 0.009310X j 0.009310X 1.097 The equatio for the dollar-weighted iterest rate, j, is: 1 0.5 0.25 X 100(1 j) 20(1 j) 40(1 j) Sice all of the cash flows occur withi a year, we ca use the simple iterest approximatio o the above equatio. We have: X 100(1 1 j) 20(1 0.5 j) 40(1 0.25 j) 100 100 j20 10 j40 10 j 120 100j Substitutig our earlier result for j ito this equatio, we ca solve for X: X 120 100(0.009310X 1.097) 0.068966X 10.3 X 149.35 ActuarialBrew.com 2013 Page 24
Solutio 34 C Derivatives ad hedgig Statemet C is the oly statemet that goes agaist the idea of usig a derivative for hedgig purposes. Hedgig reduces risk. Statemet C suggests that the derivative is used for a speculative purpose i which risk is icreased. All of the other statemets uses a derivative to lower exposure to a certai risk. Solutio 35 B Log put max gai ad loss A log put has a maximum loss of the future value of its premium ad a maximum gai of its strike price less the future value of its premium, so choice B is correct. A short put has a maximum loss as the future value of its premium less its strike price ad a maximum gai of the future value of its premium, so choice A is ot correct. A log call has a maximum loss as the future value of its premium ad a ulimited maximum gai, so choice C is ot correct. A short call has a ulimited maximum loss ad a maximum gai of the future value of its premium, so choice D is ot correct. A log collar ivolves buyig a put at a lower strike price ad sellig a call at a higher strike price. The maximum loss of a log collar combies the maximum loss of the log put (the future value of the log put premium) with the maximum loss of the short call (ulimited), so the maximum loss of a log collar is ulimited. The maximum gai of a log collar combies the maximum gai of the log put (the lower strike price less the future value of the log put premium) with the maximum gai of the short call (the future value of the call premium). Choice E is ot correct. ActuarialBrew.com 2013 Page 25